Quasi-isometries for certain right-angled Coxeter groups

We construct the JSJ tree of cylinders T₂ for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas J. Topol. 10 (2017), 1066–1106 given for certain hyperbolic RACGs. Additionally, we prove that T₂ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided K₄. By use of the structure invariant of T₂ introduced by Cashen and Martin Math. Proc. Cambridge Philos. Soc. 162 (2017), 249–291, we obtain a quasi-isometry invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry invariant in case the JSJ decomposition of the RACG does not have any rigid vertices.